Probability Theory
Probability theory, as part of analytic number theory in mathematics and science, deals with the study of probability distributions and random variables. It provides a rigorous mathematical framework for quantifying uncertainty and making predictions about uncertain events, drawing upon concepts from measure theory, topology, and real analysis. The subject encompasses various branches, such as stochastic processes, Markov chains, martingales, Brownian motion, and statistical mechanics. By exploring the convergence of sequences of random variables, probability theory also offers insights into prime number patterns and the distribution of primes in the integers, thus connecting it to analytic number theory.